Shuttleworth tension revisited

Abstract

On a solid or liquid surface in thermodynamic equilibrium, the Shuttleworth tension is a sum of two pressures (or tensions) of different nature ${\rm Y}=\gamma +\overline{{\sigma }_{xx}}$ (we consider only the diagonal component ’$xx$’). The two pressures are parallel to the surface and are practically located in the surface monolayer. The surface area is $A=N_S{A}_U^L$, $N_S$ being the number of entities in the surface monolayer and $A_U^L$ the area unit which is the Lagrangian surface area of one entity. The total energy includes the surface energetic term $\gamma A$. Its derivative with respect to $N_S$, holding $A_U^L$ constant, is the tension ${\gamma }^P$. It is numerically equal to the energy $\gamma$. The variation is of chemical nature and discontinuous. The surface monolayer has a chemical potential excess with respect to bulk for one entity ($\gamma A_U^L=\Delta \mu$). The other derivative, with respect to $A_U^L$, holding $N_S$ constant, gives the pressure of ’elastic’ nature $\overline{{\sigma }_{xx}}$. It is $\partial \gamma /\partial {ϵ}_{xx}$. For a solid, $\overline{{\sigma }_{xx}}$ decreases rapidly with the temperature, while $\gamma$ varies little. For a liquid and when neglecting $\overline{{\sigma }_{xx}}$, the literature shows that ${\rm Y}$ is reduced to the superficial tension ${\gamma }^P$. Nevertheless, our simulations on the surface $Pt\left(111\right)$ show that $\overline{{\sigma }_{xx}}$ and $\gamma$ are of the same order of magnitude for the liquid state. The Laplace force is thus proportional to the sum of two pressures when the surface is curved.

For a surface considered as flat, the surface tension is a dipole force internal to the whole system. On vicinals, the surface tension tends to contract the atoms on the terrace, but the atoms are less contracted because the step energy does not change between the two equilibrium configurations that differ only by the presence or otherwise of the surface tension. The step–step interaction stress is repulsive and varies as $1/L$, $L$ being the step–step distance. Regardless of the step energy, the step stress also contributes to the equilibrium repulsion between steps.